The objective is to determine the number of sides of two regular polygons, A and B, given that the ratio of their interior angles is 3:4. The process is divided into sequential steps:
Let n represent the number of sides of polygon A. Consequently, polygon B will have 2n sides, reflecting a 1:2 ratio in the number of sides.
The formula for the sum of interior angles in a polygon with n sides is:
\(\text{Sum of interior angles} = (n - 2) \times 180^\circ\)
For a regular polygon, the measure of each interior angle is calculated as:
\(\text{Each interior angle} = \frac{(n - 2) \times 180}{n}\)
The provided ratio of interior angles between polygons A and B is 3:4. This translates to the following equation:
\(\frac{ \frac{(n - 2) \times 180}{n} }{ \frac{(2n - 2) \times 180}{2n} } = \frac{3}{4}\)
The equation is simplified as follows:
\(\frac{ (n - 2) \times 180 }{ n } \times \frac{ 2n }{ (2n - 2) \times 180 } = \frac{3}{4}\)
Further simplification yields:
\(\frac{ (n - 2) \times 2n }{ n \times (2n - 2) } = \frac{3}{4}\)
After canceling common factors and simplifying:
\(\frac{ 2(n - 2) }{ 2n - 2 } = \frac{3}{4}\)
The value of n is found through cross-multiplication:
\(4 \times 2(n - 2) = 3 \times (2n - 2)\)
Expanding both sides results in:
\(8(n - 2) = 6n - 6\)
The simplified form is:
\(8n - 16 = 6n - 6\)
Rearranging terms to group n values:
\(8n - 6n = -6 + 16\)
\(2n = 10\)
Dividing by 2 gives the solution for n:
\(n = 5\)
Since polygon B has 2n sides, the calculation is:
\(2 \times 5 = 10\)
Polygon A possesses 5 sides, and polygon B has 10 sides.